Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms

We describe a family of finite, four-dimensional, L-loop Feynman integrals that involve weight-(L+1) hyperlogarithms integrated over (L-1)-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau manifolds. At three loops, we identify the relevant K3 explicitly and we provide strong evidence that the four-loop integral involves a Calabi-Yau threefold. These integrals are necessary for the representation of amplitudes in many theories - from massless φ4 theory to integrable theories including maximally supersymmetric Yang-Mills theory in the planar limit - a fact we demonstrate.